Okay, so my problem is, that I have difficulty visualizing graphs of functions.
For example, if we have to calculate the area bounded by multiple curves, I face difficulty in visualizing that how the ...

I'm currently reading through Griffiths and Harris Principles of Algebraic Geometry, and the only subject in the foundational material section that I am not completely comfortable with is riemannian ...

What are the most important corollaries/consequences and applications of Division Algorithm, Euclidean Algorithm and Fundamental Theorem of Arithmetic?
I've been studying Elementary Number Theory for ...

I'm studying some theory about module's length and want to know motivation for this definition. I know that it's useful for intersection theory, but i know only one example from intersection theory: ...

I don't understand why it is so difficult to find a really beginner book in Algebraic Geometry. Let's take for example Fulton's algebraic curves: An introduction to Algebraic Geometry. I've already ...

I have a bit of confusion about the following situation. Let's assume that we have a symmetric function $f(x,y)$ where it has the property $f(x,y) = f(y,x)$ for all $x$ and $y$. $x$ and $y$ have the ...

Do mathematicians do anything else beside writing proofs? It seems like all the "upper-division" math here are about proving something rather than solving for something i.e. instead solving for $x^2 = ...

I've always wanted to come up with a fairly concrete example of an object or realistic set that could be uncountable. Most of the sets I can think about, even the hugest ones, are always countable. ...

Few days ago I failed my Calculus exams. And again it was mostly due to simple mistakes such as forgetting about minus in front of fraction, switching y coordinates of two points etc.
The assignments ...

Recently (although still in high school) I've been at university, more precisely at information science engineering as apprenticeship. I want to become an operating system programmer but I severely ...

Actually, I was seeking for top universities, which has mathematics depart, in Pakistan and I found one, namely Quaid-i-Azam University. Which is known for its Education in "Natural Science". Then I ...

So I am totally new to the field of random matrices, but I was not sure about how they are applied. According to Wikipedia, a random matrix is "a matrix-valued random variable—that is, a matrix some ...

I am a second year undergraduate college student interested in applied math program. I hear a lot that general topology(e.g. the first half of Munkres' book Topology) is very useful, but is it really ...

Is Algebraic Number Theory the study of the theory of algebraic numbers? Or is it the study of the theory of numbers from an algebraic viewpoint?
Or is it both?
I know I can just find a wiki article ...

Recently I'm studing some basic homotopy theory. An important brick of the exact sequence of cofibration is the following sequence: $$X \stackrel{f}{\longrightarrow} Y \stackrel{i}{\longrightarrow} C ...

Suppose $X$ is a connected topological space. A typical way that we prove a property $P(x)$ holds for all $x \in X$ is to show that $P$ is an open and a closed condition, and that $P(x)$ for some $x ...

Some background information first of all:
I'm 16 now and just started studying mathematics intensely. I live in the UK and my goal is to eventually become very good at advanced mathematics (graduate ...

How to call the $n$ in $\mathbb{R}^{n}$ in general? It is cumbersome to say something like $n$ is the number of the folds of $\mathbb{R}$ in the Cartesian product ...
If $\mathbb{R}^{n}$ is regarded ...

I am no great mathematician but I have a question which I can't seem to find a answer for. How can one divide a finite by a infinite number? For example if you have a circle with a circumference of ...

In a post Terence Tao explained a very nice way to think about convolution and noted that "one should try to use physical intuition to model mathematical concepts whenever one can". I found this very ...

Im reading about Calculus of varations and there is a lot of references to "the functional" i.e we want to find the minimum of the functional etc. From what i have read, "the functional" is simply the ...

So, let me first describe how my doubt originated: out of curiosity I started to study Newton's Opticks, a book written more than 300 years ago. I was doing some of the experiments described on it, ...

I was told to make a report for mathematics, and I could choose my own subject. I chose complex numbers, because I really think they are interesting. However, my teacher says that there isn't a lot of ...

I know that the donut-shaped geometric object in $\mathbb{R}^3$ is homeomorphic to a square with identified opposite sites. However, while the latter has a clear symmetry between two dimensions, the ...

I am a mathematics student in my last year of undergraduate studies and I'm taking a first Course in Category Theory. The professor that is giving the course is not a category theorist and because of ...

Let's suppose that I have proved a theorem myself and I want to write an article about it, I have a few questions:
1) How do I have to write it ? I mean, what character should I use, what conventions ...

$ \left( \dfrac{\partial}{\partial x} \right)_{p} $ is both an element of the tangent space $ {T_{p}}(M) $ and a linear functional on $ {C^{1}}(M) $, while $ (\mathrm{d}{x})_{p} $ is an element of the ...

I know that the notation $\binom{n}{r}$ is more standard to use since we have a $\LaTeX$ command for it while there is no such thing for $^n\mathrm{C}_r$.
Now, I'm wondering which notation do people ...

I'm a student who has about a year (and a few months) to go before entering a university and I've been reading some math books recently. I'm on Chapter 6 on Rudin's PMA, Chapter 5 in Munkres' Analysis ...